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/C++

std::comp_ellint_1, std::comp_ellint_1f, std::comp_ellint_1l

double      comp_ellint_1( double k );
float       comp_ellint_1( float k );
long double comp_ellint_1( long double k );
float       comp_ellint_1f( float k );
long double comp_ellint_1l( long double k );

(1)

(since C++17)

double      comp_ellint_1( IntegralType k );

(2)

(since C++17)

1) Computes the complete elliptic integral of the first kind of k.

2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

Parameters

elliptic modulus or eccentricity (a value of a floating-point or integral type)

Return value

If no errors occur, value of the complete elliptic integral of the first kind of k, that is ellint_1(k,π/2), is returned.

Error handling

Errors may be reported as specified in math_errhandling.

If the argument is NaN, NaN is returned and domain error is not reported
If |k|>1, a domain error may occur

Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header tr1/cmath and namespace std::tr1.

An implementation of this function is also available in boost.math.

The period of a pendulum of length l, given acceleration due to gravity g, and initial angle θ equals 4√l/gK(sin2
(θ/2)), where K is std::comp_ellint_1.

Example

#include <cmath>
#include <iostream>
int main()
{
    double hpi = std::acos(-1)/2;
    std::cout << "K(0) = " << std::comp_ellint_1(0) << '\n'
              << "π/2 = " << hpi << '\n'
              << "K(0.5) = " << std::comp_ellint_1(0.5) << '\n'
              << "F(0.5, π/2) = " << std::ellint_1(0.5, hpi) << '\n';
    std::cout << "Period of a pendulum length 1 m at 90 degree initial angle is "
              << 4*std::sqrt(1/9.80665)*
                 std::comp_ellint_1(std::pow(std::sin(hpi/2),2)) << " s\n";
}

Output:

K(0) = 1.5708
π/2 = 1.5708
K(0.5) = 1.68575
F(0.5, π/2) = 1.68575
Period of a pendulum length 1 m at 90 degree initial angle is 2.15324 s

External links

Weisstein, Eric W. "Complete Elliptic Integral of the First Kind." From MathWorld--A Wolfram Web Resource.